GEOPHYSICAL RESEARCH LETTERS, VOL. 39, L17204, doi:10.1029/2012GL052736, 2012
Enceladus’ extreme heat flux as revealed by its relaxed craters
Michael T. Bland,1 Kelsi N. Singer,1 William B. McKinnon,1 and Paul M. Schenk2
Received 14 June 2012; revised 8 August 2012; accepted 9 August 2012; published 15 September 2012.
[1] Enceladus’ cratered terrains contain large numbers of
unusually shallow craters consistent with deformation by viscous relaxation of water ice under conditions of elevated heat
flow. Here we use high-resolution topography to measure the
relaxation fraction of craters on Enceladus far from the active
South Pole. We find that many craters are shallower than
expected, with craters as small as 2 km in diameter having
relaxation fractions in excess of 90%. These measurements
are compared with numerical simulations of crater relaxation
to constrain the minimum heat flux required to reproduce these
observations. We find that Enceladus’ nominal cold surface
temperature (70 K) and low surface gravity strongly inhibit
viscous relaxation. Under such conditions less than 3% relaxation occurs over 2 Ga even for relatively large craters (diameter 24 km) and high, constant heat fluxes (150 mW m 2).
Greater viscous relaxation occurs if the effective temperature at
the top of the lithosphere is greater than the surface temperature
due to insulating regolith and/or plume material. Even for an
effective temperature of 120 K, however, heat fluxes in excess
of 150 mW m 2 are required to produce the degree of relaxation observed. Simulations of viscous relaxation of Enceladus’
largest craters suggest that relaxation is best explained by a
relatively short-lived period of intense heating that decayed
quickly. We show that infilling of craters by plume material
cannot explain the extremely shallow craters at equatorial and
higher northern latitudes. Thus, like Enceladus’ tectonic terrains, the cratered regions of Enceladus have experienced periods of extreme heat flux. Citation: Bland, M. T., K. N. Singer,
W. B. McKinnon, and P. M. Schenk (2012), Enceladus’ extreme heat
flux as revealed by its relaxed craters, Geophys. Res. Lett., 39, L17204,
doi:10.1029/2012GL052736.
1. Introduction
[2] Despite its small size, Enceladus is currently one of the
most geologically active bodies in the Solar System. Observations of the satellite’s South Polar Terrain (SPT) by the
Cassini spacecraft have revealed an essentially crater-free (i.e.,
very young), highly tectonized surface dominated by roughlyparallel fissures dubbed “tiger stripes” [Porco et al., 2006],
which are the source of extensive plumes of gas and dust
[e.g., Spitale and Porco, 2007]. Cassini’s Composite Infrared
Spectrometer (CIRS) instrument has measured extremely high
power emission in the SPT: 15.8 3.1 GW [Howett et al.,
2011]. If averaged over the region (e.g., south of 65 S), the
1
Department of Earth and Planetary Sciences and McDonnell Center for
the Space Sciences, Washington University, Saint Louis, Missouri, USA.
2
Lunar and Planetary Institute, Houston, Texas, USA.
Corresponding author: M. T. Bland, Department of Earth and Planetary
Sciences, Washington University, Campus Box 1169, 1 Brookings
Dr., Saint Louis, MO 63130, USA. ([email protected])
©2012. American Geophysical Union. All Rights Reserved.
0094-8276/12/2012GL052736
emitted power corresponds to heat fluxes of 400 mW m 2.
High resolution CIRS data indicates, however, that the regions
of elevated heat flows are strongly associated with the tiger
stripes themselves [Spencer et al., 2006; Howett et al., 2011].
[3] In addition to the SPT, two other extensive, quasicircular regions of Enceladus’ surface have been tectonically
modified: one centered on the leading hemisphere, and one
centered on the trailing hemisphere [e.g., Porco et al., 2006;
Spencer et al., 2009; Crow-Willard and Pappalardo, 2011].
Together with the SPT, these regions cover a little more than
50% of the satellite [Crow-Willard and Pappalardo, 2011].
Numerical modeling of the tectonic surface deformation in
these regions suggest that, like the SPT, heat fluxes likely were
quite high at the time of disruption: 110-to-270 mW m 2
[Bland et al., 2007; Giese et al., 2008], larger than typical
terrestrial heat fluxes, and much larger than the roughly 1-to3 mW m 2 expected radiogenic heat flux at Enceladus. The
evidence for similar geologic disruption styles in spatially and
temporally separated regions has lead to the inference that
Enceladus undergoes quasi-episodic periods of localized high
heat flow and tectonic disruption [Helfenstein et al., 2006;
O’Neill and Nimmo, 2010].
[4] Outside of these tectonized regions, Enceladus’ surface
is dominated by cratered terrains with few tectonic features
and age estimates ranging from 1 Ga [Kirchoff and Schenk,
2009] to 2 Ga [Porco et al., 2006], assuming the updated
cometary cratering fluxes of Dones et al. [2009] (age uncertainties are large). The most notable characteristic of the
cratered terrain is the large number of impact craters displaying morphologies consistent with viscous relaxation
(e.g., shallow depths, up-domed floors) [e.g., Smith et al.,
1982; Passey, 1983]. These viscously relaxed craters are
markers of elevated past heat flow [e.g., Passey, 1983], and
therefore belie the notion that Enceladus’ cratered terrain has
been geologically quiescent throughout the satellite’s history.
[5] In this paper we utilize high-resolution topography data
produced by stereo-controlled photoclinometry to characterize
the degree of relaxation (i.e., their current apparent depth
relative to their expected initial depth) for a sub-set of craters
on Enceladus. These observations are compared to finite
element simulations of the viscoelastic relaxation of impact
craters to place quantitative constraints on the long-term heat
flux in Enceladus’ cratered terrains. We find that, even when
using conservative assumptions (i.e., assumptions that enhance
relaxation), heat fluxes in the cratered terrain must have been
>150 mW m 2. We suggest then, that even Enceladus’ endogenically inactive regions have had a spectacular (and possibly
non-uniform) thermal history.
2. Measuring Crater Relaxation on Enceladus
[6] We measured apparent crater depths (i.e., crater depth
relative to the surrounding ground plain) using ArcGIS in two
regions on Enceladus (Figure 1). Within these two regions
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Voyager 2 [Smith et al., 1982; Passey, 1983]; however, our
measurements emphasize the extreme degree of viscous
relaxation that has occurred at even small crater sizes, unresolvable in Voyager imagery.
3. Modeling Crater Relaxation
Figure 1. The two regions in which crater relaxation was
measured (outlined in yellow) (a) north of Hamah Sulci
(24 craters) and (b) east of Harran Sulci (127 craters). See
Figure S1 for context map.
data were acquired for 151 craters for which the illumination
geometry provided robust topography (127 equatorial craters, 24 craters north of Hamah Sulcus). Crater topography
was derived from high resolution Cassini ISS images using
stereo-controlled photoclinometry. The technique has been
described in detail elsewhere [see, e.g., Schenk, 2002; Bray
et al., 2012]. The combined stereo/photoclinometry technique provides a robust topographic data set at both long and
short wavelengths with vertical precisions of a few 10s of
meters. Details of the measurement techniques and the
uncertainties are provided in the auxiliary material.1
[7] From the apparent crater depths (da), the degree of
relaxation (i.e., the relaxation fraction (RF)) can be calculated
if an initial apparent crater depth (da,i) is assumed (RF =
1
da/da,i). Initial crater depths are derived from lunar
depth-diameter relations, which are applicable to simple
craters on the terrestrial planets [e.g., Melosh, 1989] and the
icy Galilean satellites [Schenk, 2002] (see auxiliary material).
The relaxation fraction for all 151 craters plotted against their
crater diameter is shown in Figure 2. In general, craters on
Enceladus are highly relaxed. All measured craters over
12 km were effectively completely relaxed (i.e., only shortwavelength topography remains). Smaller craters (<10 km)
show greater variability in their relaxation state. Even some of
the smallest craters measured (2 km diameter) have undergone more than 90% relaxation: a notable observation considering the difficulty involved in relaxing short-wavelength
topography [e.g., Scott, 1967; Parmentier and Head, 1981].
That relaxed craters exist on Enceladus has been known since
1
Auxiliary materials are available in the HTML. doi:10.1029/
2012GL052736.
[8] In order to constrain the thermal conditions required to
produce the degree of viscous relaxation observed on Enceladus we simulate crater relaxation in axisymmetric geometry
using the viscoelastic finite element code Tekton [Melosh
and Raefsky, 1980]. The code has been modified and used
extensively to model the tectonic deformation of ice lithospheres [e.g., Bland et al., 2007]. Simulations were performed following the approach of Dombard and McKinnon
[2006], and a more detailed description of our methodology
is available there and in the auxiliary material.
[9] We simulate the viscous relaxation of craters with
diameters between 4 and 34 km over timescales up to 2 Ga,
consistent with the maximum estimated surface age of the
cratered terrains. Because fresh craters are uncommon on
Enceladus [Kirchoff and Schenk, 2009] (Figure 2), and to
allow greater consistency between simulated craters, we utilized standardized, simple, parabolic initial crater shapes with
initial rim-to-floor depths given by d/D = 0.2 (where d is the
rim-to-floor depth and D is the rim-to-rim diameter), rim
heights of hrim = 0.036D, and rim topography (ejecta plus
Figure 2. Relaxation fraction as a function of crater diameter for measured Enceladus craters (black dots, n = 151) and
simulated craters at four different values of constant heat flux
(curves) after 2 Ga using (a) Enceladus’ nominal average
blackbody surface temperature of 70 K and (b) a lithospheric
surface temperature of 120 K. Relaxation fraction in the
numerical simulations is measured analogously to Enceladus’
actual craters: the average depth of the crater center (within
one-third crater radius)is averaged with the deepest point;
see auxiliary material for details.
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uplift) that falls off as 1/r3 (i.e., we follow fresh, lunar simple
crater geometry [Pike, 1977]).
[10] We investigate heat fluxes between 10 and 150 mW
m 2. Tekton does not include thermodynamics (i.e, temperature, heat flow, viscous dissipation are not solved for),
but viscosity structures that correspond to pre-defined heat
fluxes can be modeled. Enceladus’ nominal surface temperature is 70 K; however, Passey and Shoemaker [1982]
and Passey [1983] argued that the temperature at the top
of Enceladus’ ice lithosphere may exceed the surface temperature if it is overlain by a strongly insolating regolith.
Measurements of the thermal conductivity of fine-grained,
powdered water-ice samples in vacuum indicate values
several orders of magnitude lower than intact ice [Seiferlin
et al., 1996; Ross and Kargel, 1998], suggesting that a
few meters of regolith (or plume material) can strongly
modulate the temperature at the top of Enceladus’ lithosphere [see also Nimmo et al., 2003, Appendix B]. The
temperature at the base of an ice regolith is self limiting due
to annealing of ice grains [Passey and Shoemaker, 1982;
Nimmo et al., 2003]; we therefore investigate relaxation for
range of individual surface temperatures between 70 K and
120 K (near the suggested effective surface temperature of
Passey and Shoemaker [1982]).
[11] We model Enceladus’ lithosphere with a viscoelastic
rheology appropriate for ice Ih. We assume the ice lithosphere
is solid and coherent with a Young’s modulus of 9.3 GPa,
and a Poisson’s ratio of 0.325 [Gammon et al., 1983]. We
incorporate viscous deformation with a composite flow law
that includes dislocation creep, grain boundary sliding (GBS),
basal slip (BS), and diffusion creep. The GBS and diffusion
creep flow mechanisms are grain size dependent. We use a
nominal grain size of 1 mm, consistent with grain sizes in
terrestrial glaciers [e.g., De La Chappelle et al., 1998], and
theoretical estimates from icy satellite convection models
[Barr and McKinnon, 2007] (see below). At the maximum
stresses typical of our simulations (104–105 Pa), for larger
craters grain-size-sensitive creep mechanisms dominate viscous deformation [see Durham and Stern, 2001].
4. Model Results
[12] Figure 2 shows curves of relaxation fraction as a
function of crater diameter for four constant heat fluxes
applied over 2 Ga. Figure 2a demonstrates that Enceladus’
nominal cold surface temperature (70 K) and low gravitational acceleration strongly inhibit viscous relaxation. Even
under high heat fluxes (150 mW m 2), relatively large craters (24 km diameter) relax by less than 3% over 2 Ga. Small
craters (<10 km) undergo negligible relaxation even at the
highest heat fluxes investigated.
[13] The absence of viscous relaxation under Enceladuslike conditions in our simulations is consistent with Passey
[1983], who found from analytical models of Newtonian
relaxation that Enceladus’ nominal lithosphere was too viscous to allow relaxation of crater topography. Following the
suggestion of Passey [1983], we have investigated viscous
relaxation of impact craters using a warmer surface temperature boundary condition (as described above). Alternatively,
Enceladus’ lithosphere could contain some component (e.g.,
ammonia) that decreases its viscosity below that of pure
water ice; however because low-stress, grain-size-sensitive
rheologies of ammonia-water ices have not been measured
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[Durham et al., 1993], we currently only examine the role of
lithospheric surface temperature.
[14] Figure 2b is identical to Figure 2a but uses a lithospheric surface temperature of 120 K. As expected, with a
higher surface temperature (and therefore greater temperatures at depth for a given heat flow), viscous relaxation
occurs more readily. For craters larger than 16 km in
diameter, significant relaxation (≥40%) occurs when constant heat fluxes of 50 mW m 2 or more are applied over
2 Ga. These craters show the up-domed floors characteristic
of viscously relaxed craters. At very high heat flows relaxation fractions appear to decrease modestly as crater size
increases. This is due to the averaging technique used to
calculate the crater depth, and hence the relaxation fraction
as well (as described in the figure caption). At crater sizes
below 10 km, some relaxation (10–30%) occurs when the
highest heat fluxes in our study are applied (150 mW m 2).
Yet even with a lithospheric surface temperature 50 K above
Enceladus’ nominal average surface temperature and heat
fluxes that exceed the expected radiogenic heating (average
of 1 mW m 2 over the last 2 Ga) by two orders of magnitude, our simulations cannot produce the degree of relaxation actually observed for small craters.
[15] We have so far assumed constant heat fluxes sustained
over long periods of time, the fluxes above should be regarded as underestimates of peak values. In reality, a shorterlived, higher-amplitude heat pulse may be more geologically
plausible than the sustained heat fluxes examined. Evidence
for such a scenario comes from Enceladus’ morphologically
distinctive “central mound” craters Aladdin and Ali Baba.
These craters, located outside the two study regions described
above, have large central structures that extend to roughly
half the crater radius and tower to nearly 1 km above the
surrounding terrain (Figure 3).
[16] Comparison between topographic profiles across the
central mound crater Aladdin, and topography profiles produced by our viscous relaxation simulations of similarly sized
craters (Figure 3) suggest that the basic shape of the central
mound craters can be reproduced by viscous relaxation under
high heat fluxes for short periods of time (e.g., 150 mW m 2
over 2 Ma, with Ts = 120 K for the simulated crater in
Figure 3). Both simulated and real topography profiles exhibit
uplifted rims, deep circumferential troughs inside the crater
rim, and broad central structures that rise to 1 km above the
surrounding terrain. The central mound morphology shown in
Figure 3 is best matched by an initial crater with a modest
central peak, which is plausible given the simple-to-complex
morphological transition at a diameter of 25 km observed on
Mimas [Kirchoff and Schenk, 2009, Figure 9].
[17] The central mound morphology is a result of the
wavelength dependence of viscous relaxation: long topographic wavelengths (i.e., the crater bowl) relax quickly,
lifting the shorter-wavelength (and slowly relaxing) central
peak well above the surrounding plain. The morphology
shown in Figure 3 is transient. As relaxation continues the
uplifted crater floor subsides, leaving only relatively subdued, shorter-wavelength topography. Preserving a large
central mound therefore requires that the heating event that
causes relaxation end before the crater completely relaxes.
Assuming Ts = 120 K, heat fluxes of 50–100 mW m 2
reproduce the morphology, but require the heating to end by
20 Ma and 3 Ma, respectively, to prevent the crater from
becoming over-relaxed. Heat fluxes below 50 mW m 2 fail
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Figure 3. (top) Stereo-controlled photoclinometry data
overlaid on Cassini ISS image of two central mound craters
(Aladdin and Ali Baba; see Figure S1 for locations). The
horizontal white bar marks the position of the topography
profile shown below. (bottom) Topography profile across
the crater Aladdin (black line). Also shown is the initial
(red dotted line) and final (red solid line) shape of a simulated relaxed crater with D = 34 km after 2 Ma under a heat
flux of 150 mW m 2. The initial crater shape is a 4th order
polynomial with da/D consistent with lunar complex craters
from Pike [1977], but with a simple-to-complex transition
diameter of 25 km.
to reproduce the morphology due to insufficient uplift of the
crater center. Decreasing the lithospheric surface temperature
requires an increase in the heat flux; however, very cold
surface temperatures (e.g., Ts = 70 K) prevent sufficient
crater relaxation at any heat flux in the range investigated by
us. If central mound craters are produced by viscous relaxation, their presence suggests not only warm lithospheric
surface temperatures but also transient pulses of extremely
high heat fluxes, rather than sustained, lower heat fluxes.
5. Discussion
5.1. Can Plume Fallout Erase Craters on Enceladus?
[18] Crater counts on Enceladus by Kirchoff and Schenk
[2009] indicate that Enceladus is deficient in impact craters
greater than 6 km in diameter relative to the other Saturnian
satellites. This dearth of impact craters has been attributed
to a combination of viscous relaxation and burial by plume or
E-ring material [Kirchoff and Schenk, 2009]. Our modeling
indicates that viscous relaxation alone probably cannot
completely remove (i.e., “erase”) the topography of even
Enceladus’ larger (20 km diameter) impact craters, for “reasonable” heat flows. From these simulations, however, we
can place bounds on the volume of plume material required
to completely bury the remaining post-relaxation crater.
[19] A 20-km-diameter crater, submitted to a constant heat
flux of 150 mW m 2 for 1 Ga with a lithospheric surface
temperature of 120 K will relax by 87% as measured from
the average floor depth (Figure 2) or 82% as measured to
the deepest point in the crater. Assuming an initial true crater
depth of 4 km (i.e., rim-to-floor depth given by d/D = 0.2)
and a rim height one-fifth the total depth, a crater with an
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average and maximum apparent depth of 420 m and 580 m
will remain. Current maximum deposition rates in the equatorial region are 10 3 mm yr 1 (decreasing to the north and
away from the 45 /225 W longitude band) [Kempf et al.,
2010], requiring at least 500 Ma of sustained deposition to fill the crater. If the deposition rate was lower (say,
10 4 mm yr 1), or active only episodically (say, 10% of
the time), the time scale for crater erasure exceeds the age of
the Solar System. For smaller craters, the degree of viscous
relaxation resulting from a sustained heat flux is substantially
less, but their shallower depth requires less material to fill.
Under the conditions described above, an 8 km-diameter
crater with an initial apparent depth of 1.28 km relaxes by a
maximum of 22% leaving a crater nearly 1 km deep. As in the
case of the 20 km crater, infilling by plume material could
plausibly erase such a crater over a billion year timescales, but
only assuming high, and sustained, deposition rates. Notably,
deposition of plume material is negligible in our mid-latitude
study area north of Hamah Sulcus [Kempf et al., 2010].
[20] Fundamentally, the complete removal of craters by
relaxation and plume burial alone is problematic under the
conditions investigated here because too little viscous relaxation occurs under even large, sustained heat fluxes. If small
to mid-sized craters have been removed from the surface,
heat fluxes must have been well in excess of 150 mW m 2 or
the rate of material deposition must have been substantially
higher. The latter may require that the locus of plume activity
has shifted throughout Enceladus’ history [e.g., Helfenstein
et al., 2006; Spencer et al., 2009], permitting significantly
higher deposition rates of plume material near the cratered
terrains (e.g., deposition rates within the SPT can exceed
10 2 mm yr 1). Impact erosion and regolith formation also
act to reduce crater topography, but not in a way that mimics
relaxation. Alternatively, very small ice grain sizes (100
mm), brittle failure viscosity-reducing ice contaminants, or
even warmer effective surface temperatures could all promote greater viscous relaxation; however, the plausibility of
the first of these is questionable, and substantially relaxing
small craters will remain challenging without high heat flow
(see auxiliary material).
5.2. Enceladus’ Thermal History
[21] The high heat fluxes inferred for Enceladus’ heavily
cratered terrain alters common conceptions of the satellite’s
geologic history. Several authors have posited the idea that
Enceladus has experienced episodic periods of spatially
localized high heat fluxes and tectonic disruption. Whereas
such localized tectonic disruptions clearly have occurred, the
results presented here indicate that (1) periods of high heat
flux have occurred on Enceladus without being accompanied
by faulting, folding or similar tectonic deformation, and/or
(2) that regions of high heat fluxes were not particularly
localized.
[22] Constraining the relative timing of the heating event(s)
that caused viscously relaxation of Enceladus’ craters is difficult. Since most mid-sized (10-to-30 km diameter) craters
are at least partially relaxed, however, the high heat fluxes
must have persisted into geologically recent times (or else
numerous mid-sized unrelaxed craters should be observed).
While the cratered terrains are significantly older than
Enceladus’ tectonic terrains, the viscous relaxation event(s)
plausibly could have occurred during the same epoch of high
heat fluxes that led to the formation of the equatorial tectonic
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regions, which lie in close proximity to the cratered terrains.
Crater counts suggest that these tectonic regions have variable ages from 10 to 170 Ma [Porco et al., 2006; Kirchoff
and Schenk, 2009]. Thus, these tectonic regions could have
formed concurrently with viscous relaxation in cratered terrains, with spatial variations in heat fluxes or local stress
fields producing tectonic disruption in some areas, but only
viscous relaxation in others. Whatever the scenario, we
emphasize that the heat fluxes that have been inferred here
for Enceladus’ cratered terrain (150 mW m 2) are consistent
with heat fluxes inferred for its tectonic regions (110–
270 mW m 2 [Bland et al., 2007; Giese et al., 2008]), and are
of the same order (though lower by a factor of 2 or 3) as the
measured values spatially averaged over the SPT. Clearly, the
entire satellite has experienced a spectacular, and highly
variable thermal history.
[23] Acknowledgments. This work was supported by NASA’s Cassini
Data Analysis Program grant NNX11AK76G, and by a NESS fellowship to
K.N.S.
[24] The Editor thanks John Spencer and William B. Durham for their
assistance in evaluating this paper.
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